Drag
functions

What is a drag function? Ballistic
coefficient?

Drag
functions are usually the basis for computations of the deacceleration
of a
bullet due to air drag. The origin of drag functions can either be
measurements, calculations or both. Anyhow, to compute properties of
bullet trajectories one needs a theory. This theory takes measurements
or other theory as input and produces abstract concepts like drag
functions, trajectories and drop. The purpose of drag functions is to
simplify the computations. As it turns out, a single drag function is
usually sufficient to calculate rather accurately - at least for target
shooting and hunting - the properties of the bullets path for a whole
range of similar shaped bullets. I future, however, bullet
manufacturers will supply for each specific bullet its own drag
function (or other inputs). One reason for this is that radar
measurements of the bullets path and velocity are well within the
financial possibilities for the larger bullet smiths. The other reason is that the
computations can now be done on devices as common as a cell phone.On
the site of Lapua, for
instance, one can obtain specific drag functions of their bullets: download Lapua drag tables. BfX supports the most common drag tables as well accepts user supplied ones.

A
ballistic coefficient, specific for a
bullet, is used in combination with a certain drag function to compute
bullet specific properties in specific atmospheric conditons. Wind
velocity and air density are the most important ones. Wind changes the
velocity of the bullet in magnitude and direction, as does air density;
the larger the air
density, the larger the deacceleration of the bullet, the longer the
flight times. And if the flight time increases then wind changes the
trajectory more and gravity pulls on the bullet a longer time. The
bullet drops further. As various bullet
smiths specify their ballistic coefficients for different atmospheric
conditions one can easily make (small) errors if one does modify the
ballistic coefficient to match the actual atmospheric conditions. BfX
does its computations under ICAO standard atmospheric conditions.
BfX_C is used to adapt the calculations to the actual atmospheric
conditions. The figure below shows an excerpt from the Getting Started
workbook (that you can download here).

BfX
supports various drag functions for several families of bullet shapes.
The default one, named GP, is computed from deacceleration
formula's listed in the book of Arthur Pejsa (GP is shown in the figure
below). This computation involves density, which Pejsa did not mention
explicitly. However, he gave a clue in his book, on page 76 (yes it was
a kind of detective work), saying that
he adjusted his A function to equal the Mayevski A function at 2600 fps. This
means that he used the same conditions as those for which Mayevski and
Ingalls created their deacceleration parametrization. And according to
de basic programs of McCoy (yes, even more detective work) that is 60F,
30inchHg and 67% humidity. BfX contains also the drag function which
one can compute from Mayevski and Ingalls deacceleration formula's. It
is labeled GIM in the figure below. The GP, GIM and G1 drag functions
should be used with the G1 ballistic coefficients. In BfX you can obtain
drag function values via
the
function BfX_Cd. However, for all practical purposes you do not need
these and you do not have to compute air drag and other ballistic
properties yourself. BfX provides them on an easy to use way. BfX works in both super and subsonic velocity
regions (thus
above and below mach 1, or about 330 m/s) for all drag functions.

The Pejsa drag
function is suited, as is the G7 drag function, for modern match
bullets.
For most sport shooting purposes,
the drag function of Pejsa (GP), the G1 and G7 drag functions
perform good enough, see the elaboration in the "Getting Started"
workbook that you can download on this page.
Due to the inconsistencies in the specifications of the ballictic
coefficients published by many bullet manufacturers, small errors can be
introduced in your calculations. One should consider using the G7 (or
G1) ballistic coefficients from the book "Applied
ballistics for longe range shooting" in combination with the BfX G7 (or GP) drag function. These are all measured on a consistent way. The
author of the book, Brian Litz,
advocates the use of the G7 drag functions and related coefficients in
combination with modern low drag bullets.

BfX supplies an air density calculator (use in Excel the function =BfX_AD(temperature, pressure, humidity)), based on recent scientific research. With this function you can check if densities presented by others are correct.

Notice
that in the figure above the GP drag function crosses the GIM one at
2600 fps (or 792,48 m/s) just where Pejsa designed that his A
(deacceleration) function (on which GP is based) crossed the A function
of Mayevski Ingalls (of which GIM is derived).

BfX contains also a drag function
(A in the figure above, the drag is constant, Cd=0,2!) freqently used
by airgun
shooters and which is embodied in the program ChairGunPro
from Hawke Sport Optics. Hence, ballistic coefficients from
the extensive pellet databases the airgun shooters have been building
can be used with BfX.

User supplied drag functions

BfX, see the GettingStarted workbook that you can download from this
site, accepts also user supplied drag tables, eg ones that are obtained
from Lapua's site: download Lapua drag tables.
You can open them with a text editor and cut and paste the values in
Excel. Below a comparision between Lapua's QuickTarget and BfX, both
using a Lapua drag function.